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Constant Magnetic Field, Magnetics

Problem 3.237

A very long straight solenoid has a cross-section radius RR and nn turns per unit length. A direct current II flows through the solenoid. Suppose that xx is the distance from the end of the solenoid, measured along its axis. Find: (a) the magnetic induction BB on the axis as a function of xx; draw an approximate plot of BB vs the ratio x/Rx / R; (b) the distance x0x_{0} to the point on the axis at which the value of BB differs by η=1%\eta=1 \% from that in the middle section of the solenoid

Reveal Answer
3.237. (a) B=1/2μ0nI(1−x/x2+R2),B=1 / 2 \mu_{0} n I\left(1-x / \sqrt{x^{2}+R^{2}}\right), where x>0x>0 outside the solenoid and x<0x<0 inside the solenoid; see Fig. 23;23 ; (b) x0=x_{0}= =R(1−2η)/2η(1−η)≈5R=R(1-2 \eta) / 2 \sqrt{\eta(1-\eta)} \approx 5 R
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